Global weak solutions for a viscous liquid-gas model with singular pressure law

We study a viscous two-phase liquid-gas model relevant for well and pipe flow modelling. The gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid leading to a pressure law which becomes singular when transition to single-phase liquid flow occurs. In order to handle this difficulty we reformulate the model in terms of Lagrangian variables and study the model in a free-boundary setting where the gas and liquid mass are of compact support initially and discontinuous at the boundaries. Then, by applying an appropriate variable transformation, point-wise control on masses can be obtained which guarantees that no single-phase regions will occur when the initial state represents a true mixture of both phases. This paves the way for deriving a global existence result for a class of weak solutions. The result requires that the viscous coefficient depends on the volume fraction in an appropriate manner. By assuming more regularity of the initial fluid velocity a uniqueness result is obtained for an appropriate (smaller) class of weak solutions.

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